Explicit Descent via 4-Isogeny on an Elliptic Curve

Details Explicit Descent via 4-Isogeny on an Elliptic Curve Explicit Descent via 4-Isogeny on an Elliptic Curve

2004-11-09

arxiv.org/abs/math/0411215v1

Mathematics

Number Theory

20 pages

Scientific paper

We work out the complete descent via 4-isogeny for a family of rational
elliptic curves with a rational point of order 4; such a family is of the form
$y^2 + x y + a y = x^3 + a x^2$ where $\sqrt{-a} \in \mathbb Q^\times$. In the
process we exhibit the 4-isogeny and the isogenous curve, explicitly present
the principal homogeneous spaces, and discuss examples by computing the rank.

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